Lots of good covers already mentioned, but here's a couple more:
Siouxsie and the Banshees: Trust In Me (Sherman/Sherman, from the movie Jungle Book)
Ramones: I Don't Wanna Grow Up (Tom Waits)
http://www.youtube.com/watch?v=kU2SRNU955c
Most people propably don't realize/remember this is a cover:
The Doors: Alabama Song (Hauptmann/Weill, from a 1920's Bertolt Brecht play)
http://www.youtube.com/watch?v=YdPoqfcB3Gk
Scissor Sisters: Comfortably Numb (Pink Floyd)
http://www.youtube.com/watch?v=jRk5Gjl_cAg
Moloko: Are Friends Electric? (Tubeway Army (Gary Numan))
http://www.youtube.com/watch?v=Lv5xsA9gv-8
Meshell Ndegeocello: Who Is He and What is He to You (Bill Withers)
http://www.youtube.com/watch?v=K0ov9082a1c
Some not necessarily as good IMO, but worth mentioning:
HIM: Wicked Game (Chris Isaak)
http://www.youtube.com/watch?v=4WA2jBMk-Pk
Pain: Eleanor Rigby (The Beatles)
http://www.youtube.com/watch?v=eB_3JIN5Vso
Apocalyptica: Refuse/Resist (Sepultura)
http://www.youtube.com/watch?v=ZSDMCXp0xlg
Nightwish: Phantom of the Opera (Hart/Stilgoe/Batt and obviously Andrew Lloyd Webber)
http://www.youtube.com/watch?v=YJragdB6L4Q
White Stripes: Jolene (Dolly Parton)
http://www.youtube.com/watch?v=yXlULkwhgrc
Finally, I would like to advertise
my favourite cover albums ever. Mick Harvey (of Bad Seeds fame) has done two albums of Serge Gainsbourg covers, called 'Intoxitaced Man' and 'Pink Elephants'. Brilliant stuff.
Mick Harvey with Anita Lane: Harley Davidson (Serge Gainsbourg)
Mick Harvey: Pink Elephants (Serge Gainsbourg)